Schmid Law Research Articles

Deformation behaviour of nano-twinned single crystals of an Fe–19Cr–16Ni austenitic alloy

The deformation behaviour of an Fe–19Cr–16Ni austenitic alloy was studied by micro-tensile testing of small-scale single crystals. A detailed comparison of specimens with and without nano-twins was conducted. The study revealed that the deformation behaviour depends strongly on the crystallographic orientation of the parent crystal and the spacing between nano-twin bundles. Although the yield stress in single crystals without nano-twins was governed by Schmid's law, the slip transfer inhibition by twin boundaries contributed significantly to the strengthening in nano-twinned samples. With the parent crystal oriented favourably for single-slip, the dislocations in the parent crystal glided by cross-slip onto the primary plane in the twinned crystal. For the hard-oriented parent crystal with dense nano-twins, dislocation accumulated and led to necking at an early stage. For the same orientation but with sparse nano-twins, twin growth and nucleation was promoted by dislocation reactions with the twin boundaries. These findings suggest that control of the nano-twin structure can allow the desired deformation behaviour to be tuned.

Microscopic deformation compatibility during monotonic loading in a Mg-Gd-Y alloy

Microscopic deformation compatibility is a key in achieving high ductility in polycrystalline magnesium alloys. This paper presents a grain-level compatibility investigation of Mg-Gd-Y alloy using in-situ tension in scanning electron microscopy (SEM) combined with electron back scattered diffraction (EBSD) and digital image correlation (DIC) techniques. The results show that strain amplitudes are not always higher in grains with high Schmid factors of basal slip. Thus, the Schmid law can hardly predict the strain amplitudes at grain level. In addition, the average intergranular strains are higher than the average intragranular strain and a strong surrounding effect grain-to-grain was found. A slip-transfer parameter m′, describing the surrounding effect, was used to evaluate the deformation compatibility. It is found that a high m′ value and a high Schmid factor of certain slip system in the adjacent grains would result in a good compatibility around grain boundaries and could accommodate higher local strains, which would improve the ductility of magnesium alloys.

Orientation dependence of void growth at triple junction of grain boundaries in nanoscale tricrystal nickel film subjected to uniaxial tensile loading

Molecular dynamics simulation was performed in order to investigate the dependence of void growth on crystallographic orientation at the triple junction of grain boundaries in nanoscale tricrystal nickel film subjected to uniaxial tensile loading. The nucleation, the emission and the transmission of Shockley partial dislocations play a predominant role in the growth of void at the triple junction of grain boundaries. The orientation factors of various slip systems are calculated according to Schmid law. The slip systems activated in a grain of tricrystal nickel film basically conform to Schmid law which is completely suitable for a single crystal. The activated slip systems play an important role in plastic deformation of nanoscale tricrystal nickel film subjected to uniaxial tensile loading. The slip directions exhibit great difference among the activated slip systems such that the void is caused to be subjected to various stress conditions, which further leads to the difference in void growth among the tricrystal nickel films with different orientation distributions. It can be concluded that the grain orientation distribution has a significant influence on void growth at the triple junction of grain boundaries.

Plastic anisotropy and dislocation trajectory in BCC metals

Plasticity in body-centred cubic (BCC) metals at low temperatures is atypical, marked in particular by an anisotropic elastic limit in clear violation of the famous Schmid law applicable to most other metals. This effect is known to originate from the behaviour of the screw dislocations; however, the underlying physics has so far remained insufficiently understood to predict plastic anisotropy without adjustable parameters. Here we show that deviations from the Schmid law can be quantified from the deviations of the screw dislocation trajectory away from a straight path between equilibrium configurations, a consequence of the asymmetrical and metal-dependent potential energy landscape of the dislocation. We propose a modified parameter-free Schmid law, based on a projection of the applied stress on the curved trajectory, which compares well with experimental variations and first-principles calculations of the dislocation Peierls stress as a function of crystal orientation.

Anisotropy in twinning characteristics and texture evolution of rolling textured high purity alpha phase titanium

Types of primary and secondary twins together with their variant selection during uniaxial compression of rolling textured high purity alpha phase titanium were significantly dependent on the loading direction. Such an anisotropy in twinning characteristics consequently led to anisotropic texture evolution. Our results showed that the combined effect of the sense of twinning shear and Schmid factor determines types of active primary and secondary twinning systems, and the Schmid law governs their variant selection, thereby controlling texture evolution.

Observation and analysis of the coexistence of two “opposite” twin modes in a Mg-Al-Zn alloy

Understanding twinning mechanism is crucial to effectively tailor twin-related properties of HCP metals. Here, we reported an uncommon twin phenomenon during rolling of tilted Mg alloy sheets: coexistence of {10−12} extension twin (ET) and {10−11} contraction twin (CT) inside the same grain. 26 such twinning grains containing 40 ETs and 33 CTs were identified via crystallographic analysis to explore the coexistence mechanism in terms of Schmid law and local strain accommodation effects. The statistical results show that the geometrical relationship between stress state during rolling and grain orientations is favorable for the activation of CTs but not for ETs. Further examination indicates that ETs are very likely to be induced by pile-up of basal slip in their respective parent grains.

Ti–4.5Al–2Mo–1.6V–0.5Fe–0.3Si合金の強度–延性バランスに及ぼす組織因子と変形機構

The effects of difference in morphology, grain size and β fraction on tensile properties in Ti–4.5Al–2Mo–1.6 V–0.5Fe–0.3Si (Ti-9) alloy were quantitatively examined. Excellent balance of strength and ductility is exhibited in the bimodal morphology. Refinement of α-grain size results in simultaneous increase in strength and ductility. Herein, the Hall–Petch coefficients in the equiaxed morphology and bimodal morphology exhibit similar value of approximately 230 MPa/(µm−1/2), implying that the similar Hall–Petch effect is acted for both morphologies. During deformation, twinning in the α phase does not activate. On the other hand, according to the experimental results on strength and the Schmid law, we can speculate that 〈c+a〉 dislocation slip activates easier than the case in the Ti–6Al–4V alloy. In Ti-9 alloy, it is supposed that suppression of planar-dislocation-motion (under deformation) due to the decrease in Al content and the accommodation of strain incompatibility at α/β interface contribute to improve the ductility.

Crystallographic analysis on the activation of multiple twins in rolled AZ31 Mg alloy sheets during uniaxial and plane strain compression

It is well known that the activation and interception of multiple twins in Mg alloys largely influence microstructural evolution and strain hardening behavior. However, the crystallographic conditions for the activation of multiple twins and their variant selection have not been well understood. Therefore, the present work was conducted in attempt to find some trends such as grain orientation, Schmid factor and geometric compatibility factor on the activation and variant selection of multiple twins in both uniaxial compression (UC) and plane strain compression (PSC). The results showed that most of the twins in single twin grains strictly followed Schmid law. For the activation of multiple twin variants inside grains and cross-boundary twins, the local strain compatibility between the connected twins was also important for variant selection. Because of the constraint along the c-axis, twinning activation was suppressed in PSC compared to UC. In general, the variant selection of twinning was more likely affected by local strain compatibility in PSC than that in UC. These experimental results may shed light on the development of advanced physics-based simulation models.

The role of double twinning on transgranular fracture in magnesium AZ61 in a localized stress field

The current work combines electron backscattered diffraction and Schmid's law numerical analysis to study the micro-mechanisms responsible for transgranular fracture in magnesium AZ61 subjected to a triaxial stress field operating at the vicinity of a notch. It is argued that the susceptibility of an individual grain to transgranular cracking depends strongly on its orientation relative to the stress condition. Grains in which the stress field creates high shear stresses on their {1 0 1¯ 1} twinning systems are more susceptible to transgranular fracture. The primary {1 0 1¯ 1} twinning mode was determined to follow a Schmid-type behavior in most of the studied grains. An inspection of the misorientation boundaries between the parent and twinned segments reveals the role of {1 0 1¯ 1}–{1 0 1¯ 2} double twinning, which essentially involves the rotation of 12¯ 1 0/37.5° relative to the parent, on the formation of transgranular micro-cracks. This twinning mode intensifies the shear stresses resolved on the basal slip systems, leading to early shear localization and transgranular void/crack initiation.

Local deformation mechanisms of two-phase Ti alloy

This paper describes a study of local deformation mechanisms in two-phase Ti alloy, Ti–6Al–2Sn–4Zr–2Mo, by performing in-situ micropillar compression tests. A colony microstructure was examined and select grains identified for examination were chosen with EBSD measurements. These grains were chosen to isolate individual slip systems within each test. Micropillars of tri-crystal (α–β–α) structure were fabricated from four determined regions, and compression tests were performed using a displacement-controlled nanoindenter set inside a SEM, with a constant displacement rate. The results show that the α/β morphology significantly affects the local deformation behaviour. For these colony structures, Schmid's law in general enables anticipation of local slip activity, but the presence and morphology of the β phase can significantly alter the apparent yielding point and work hardening response. The role of interfaces within these tri-crystal pillars is discussed.

Predicting Texture Evolution in Ta and Ta-10W Alloys Using Polycrystal Plasticity

We present results of texture characterization and predictions of a multiscale physically based constitutive law developed to predict the mechanical response and texture evolution of body-centered cubic metals. The model is unique in the sense that single crystal deformation results not only from the resolved shear stress along the direction of slip (Schmid law) but also from shear stresses resolved along directions orthogonal to the slip direction as well as the three normal stress components (non-Schmid effects). The single crystal model is implemented into a visco-plastic self-consistent homogenization scheme containing a hardening law for crystallographic slip. The polycrystal model is calibrated using a set of mechanical test data collected on a tantalum-tungsten alloy, Ta-10W, in tension and compression and pure tantalum, Ta, in tension, compression, and cross-rolling. We demonstrate that the model effectively captures the texture evolution in all cases. We show that alloying has the effect of increasing the dislocation friction stress, the trapping rate of dislocations, and activation barrier for recovery.

Single crystal plastic behavior of a single-phase, face-center-cubic-structured, equiatomic FeNiCrCo alloy

To understand the fundamental deformation mechanisms of compositionally complex alloys, single crystals of a multi-component equiatomic FeNiCoCr alloy with face-center-cubic (FCC) structure were grown for mechanical studies. Similar to typical FCC pure metals, slip trace analyses indicate that dislocation slips take place on (111) planes along [11¯0] directions. The critical resolved shear stress (CRSS) obeys the Schmid law at both 77 and 293K, and tension–compression asymmetry is not observed. Although this material slips in a normal FCC manner both at 293 and 77K, the strong temperature dependence of the CRSS is abnormal in comparison to the typical FCC metals.

A Comparative Study of Forming Limit Diagrams Predicted by Two Different Plasticity Theories Involving Vertex Effects

The main objective of this contribution is to compare the Forming Limit Diagrams (FLDs) predicted by the use of two different vertex theories. The first theory is micromechanical and is based on the use of the Schmid law, within the framework of crystal plasticity coupled with the Taylor scale-transition scheme. The second theory is phenomenological and is based on the deformation theory of plasticity. For both theories, the mechanical behavior is formulated in the finite strain framework and is assumed to be isotropic and rate-independent. The theoretical framework of these approaches will be presented in details. In the micro-macro modeling, the isotropy is ensured by considering an isotropic initial texture. In the phenomenological modeling, the material parameters are identified on the basis of micro-macro simulations of tensile tests.

Texture evolution of rolled Mg–3Al–1Zn alloy undergoing a {10-12} twinning dominant strain path change

The texture modification, caused by {10-12}–{10-12} double twinning, in rolled Mg–3Al–1Zn alloy was investigated by carrying out consecutive in-plane compressions along two orthogonal directions in combination with an electron backscatter diffraction measurement and a Schmid factor analysis. The results revealed that the Schmid law still governs twin variant selection, thereby controlling texture evolution, however, the specific crystallographic feature of {10-12} twinned region in which the c-axis orients to the compression direction with a variation of ±30° on the rolling plane and one prismatic plane including the rotation axis of {10-12} twinning, corresponding to an a-axis, is aligned almost parallel to the rolling plane gives rise to a significantly different texture evolution of {10-12}–{10-12} double twinned region as compared to {10-12} twinned region.

A multi-scale model of dislocation plasticity in α-Fe: Incorporating temperature, strain rate and non-Schmid effects

In this work, we develop an atomistically informed crystal plasticity finite element (CP-FE) model for body-centered-cubic (BCC) α-Fe that incorporates non-Schmid stress dependent slip with temperature and strain rate effects. Based on recent insights obtained from atomistic simulations, we propose a new constitutive model that combines a generalized non-Schmid yield law with aspects from a line tension (LT) model for describing activation enthalpy required for the motion of dislocation kinks. Atomistic calculations are conducted to quantify the non-Schmid effects while both experimental data and atomistic simulations are used to assess the temperature and strain rate effects. The parameterized constitutive equation is implemented into a BCC CP-FE model to simulate plastic deformation of single and polycrystalline Fe which is compared with experimental data from the literature. This direct comparison demonstrates that the atomistically informed model accurately captures the effects of crystal orientation, temperature and strain rate on the flow behavior of siangle crystal Fe. Furthermore, our proposed CP-FE model exhibits temperature and strain rate dependent flow and yield surfaces in polycrystalline Fe that deviate from conventional CP-FE models based on Schmid's law.

Anisotropic distribution of residual strain around conical nanoindentation in silicon

Residual strain around conical nanoindentations on silicon single crystals is mapped by electronic backscatter diffraction system and CrossCourt software. Both of the (001) and (111) planes display anisotropic strain features adjacent to nanoindentations in specific crystallographic orientations with strain resolutions of 3.5×10−4 and 3.3×10−4 for Si(001) and (111) surfaces, respectively. The anisotropic distributions of most strain components induced by isotropic load present central symmetry, which are related to the crystallographic orientation and slip system in silicon microstructures according to the Schmid law. The indentation deformation of single-crystal Si is a process of dislocation slip on specific plane in specific direction.

An approximate yield criterion for porous single crystals

This study is devoted to the derivation of an approximate yield function for voided single crystals deforming by crystallographic slip . By making use of a regularized form of the Schmid law and solving approximately the hydrostatic case with a limit-analysis calculation, a Gurson-type yield criterion is proposed. It is obtained by a heuristic extension of existing limit-analysis results for a matrix obeying a quadratic Hill-type criterion. The proposed yield function is successfully compared with numerical results from the literature for face-centered cubic crystals. • An approximate plastic yield criterion is presented for voided FCC crystals • By making use of a regularized Schmid law, a Gurson-type yield function is proposed. • The limit-analysis study of a hollow sphere is reported for hydrostatic loading. • The criterion is assessed with Han et al. (2013) unit-cell FE results.

Effects of misorientation and inclination on mechanical response of 〈1 1 0〉 tilt grain boundaries in α-Fe to external stresses

Due to the industrial importance of α-iron-based polycrystalline materials, their grain boundary (GB) structures and properties need to be well characterized and understood in order to optimize the materials through effective GB engineering. In this study, a molecular dynamics (MD) simulation study was performed to investigate a series of 〈1 1 0〉 symmetric tilt grain boundaries (STGBs) and asymmetric tilt grain boundaries (ATGBs) in α-iron. It is shown that the GB energy is proportional to the GB volumetric expansion. During uniaxial deformation, 〈1 1 1〉{1 1 2} twinning appears to be more competitive or easier than 〈1 1 1〉{1 1 2} dislocation emission from the GB at yielding. For bicrystal systems containing STGBs the yield strength obeys the Schmid law, while for ATGB bicrystal systems the yield strengths are mainly determined by the local stress rather than overall stress and average GB energy. The higher degree of atomic disordering in the ATGB regions generates larger local stress fluctuation and thus facilitates local defect emission when subjected to external stresses.

Analysis of Crystallographic Orientation Changes during Deformation of Magnesium Single Crystals

Magnesium single crystals of three di erent orientations were deformed by tension at room temperature to investigate the geometrical criterion resulting from the Schmid law for activation of basal and non-basal slip systems. Changes of crystallographic orientations of investigated single crystals were systematically measured during deformation and the geometrical criterion was examined in respect to the changes of crystallographic orientations during deformation. The geometrical criterion itself failed to explain lack of activity of non-basal slip systems in the deformed magnesium single crystals. Therefore the ratio of activation volumes of soft to hard slip systems is considered to be introduced for a better understanding of the deformation in hcp metals.

A strain-rate and temperature dependent constitutive model for BCC metals incorporating non-Schmid effects: Application to tantalum–tungsten alloys

In this work, we present a multiscale physically based constitutive law for predicting the mechanical response and texture evolution of body-centered cubic (BCC) metals as a function of strain-rate and temperature. In the model, deformation of individual single crystals results not only from the resolved shear stress along the direction of slip (Schmid law) but also from shear stresses resolved along directions orthogonal to the slip direction as well as the three normal stress components (non-Schmid effects). We account for coupled Schmid and non-Schmid effects through the modification of the resolved shear stress for both 1/2〈11¯1〉{110} and 1/2〈111¯〉{112} slip systems and the modification of the slip resistance for 1/2〈111¯〉{112} slip systems. The single crystal model is implemented into a self-consistent homogenization scheme containing a hardening law for crystallographic slip. The hardening law is based on the evolution of dislocation densities that incorporates strain-rate and temperature effects through the Peierls stress, thermally activated recovery, dislocation substructure formation and dislocation interactions. The polycrystal model is calibrated and validated using a set of mechanical and texture data collected on a tantalum–tungsten alloy, Ta–10W, at temperatures ranging from 298K to 673K and strain-rates from 10−3s−1 to 2400s−1. We show the model effectively captures the anisotropic hardening rate and texture evolution for all data using a single set of single-crystal hardening parameters. Comparisons between predictions and measured data allow us to discuss the role of slip on {110} and {112} in determining plasticity and texture evolution in Ta–10W.